第9章数学モデルと幾何

What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle in Figure 9.9 is called $\text{Δ}ABC,$ read ‘triangle $\text{ABC}$’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.

The three angles of a triangle are related in a special way. The sum of their measures is $\text{180°}.$

Some triangles have special names. We will look first at the right triangle. A right triangle has one $\text{90°}$ angle, which is often marked with the symbol shown in Figure 9.10.

If we know that a triangle is a right triangle, we know that one angle measures $\text{90°}$ so we only need the measure of one of the other angles in order to determine the measure of the third angle.

In the examples so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. So we will wait to draw the figure until we write expressions for all the angles we are looking for.

When we use a map to plan a trip, a sketch to build a bookcase, or a pattern to sew a dress, we are working with similar figures. In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures. One is a scale model of the other. The corresponding sides of the two figures have the same ratio, and all their corresponding angles have the same measures.

The two triangles in Figure 9.11 are similar. Each side of $\text{Δ}ABC$ is four times the length of the corresponding side of $\text{Δ}XYZ$ and their corresponding angles have equal measures.

The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in $\text{Δ}ABC\text{:}$

We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.